Theorem sumeq2w 15617

Description: Equality theorem for sum, when the class expressions 𝐵 and 𝐶 are equal everywhere. Proved using only Extensionality. (Contributed by Mario Carneiro, 24-Jun-2014.) (Revised by Mario Carneiro, 13-Jun-2019.)

Assertion

Ref	Expression
sumeq2w	⊢ (∀𝑘 𝐵 = 𝐶 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶)

Proof of Theorem sumeq2w

Dummy variables 𝑓 𝑚 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq2 3853	. . . . . . . . . 10 ⊢ (∀𝑘 𝐵 = 𝐶 → ⦋𝑛 / 𝑘⦌𝐵 = ⦋𝑛 / 𝑘⦌𝐶)
2	1	ifeq1d 4498	. . . . . . . . 9 ⊢ (∀𝑘 𝐵 = 𝐶 → if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0) = if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))
3	2	mpteq2dv 5191	. . . . . . . 8 ⊢ (∀𝑘 𝐵 = 𝐶 → (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0)))
4	3	seqeq3d 13934	. . . . . . 7 ⊢ (∀𝑘 𝐵 = 𝐶 → seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) = seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))))
5	4	breq1d 5107	. . . . . 6 ⊢ (∀𝑘 𝐵 = 𝐶 → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥 ↔ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥))
6	5	anbi2d 631	. . . . 5 ⊢ (∀𝑘 𝐵 = 𝐶 → ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥)))
7	6	rexbidv 3159	. . . 4 ⊢ (∀𝑘 𝐵 = 𝐶 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥)))
8		csbeq2 3853	. . . . . . . . . . 11 ⊢ (∀𝑘 𝐵 = 𝐶 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑛) / 𝑘⦌𝐶)
9	8	mpteq2dv 5191	. . . . . . . . . 10 ⊢ (∀𝑘 𝐵 = 𝐶 → (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵) = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))
10	9	seqeq3d 13934	. . . . . . . . 9 ⊢ (∀𝑘 𝐵 = 𝐶 → seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)) = seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶)))
11	10	fveq1d 6835	. . . . . . . 8 ⊢ (∀𝑘 𝐵 = 𝐶 → (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))
12	11	eqeq2d 2746	. . . . . . 7 ⊢ (∀𝑘 𝐵 = 𝐶 → (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚) ↔ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚)))
13	12	anbi2d 631	. . . . . 6 ⊢ (∀𝑘 𝐵 = 𝐶 → ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))))
14	13	exbidv 1923	. . . . 5 ⊢ (∀𝑘 𝐵 = 𝐶 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))))
15	14	rexbidv 3159	. . . 4 ⊢ (∀𝑘 𝐵 = 𝐶 → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))))
16	7, 15	orbi12d 919	. . 3 ⊢ (∀𝑘 𝐵 = 𝐶 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚)))))
17	16	iotabidv 6475	. 2 ⊢ (∀𝑘 𝐵 = 𝐶 → (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))) = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚)))))
18		df-sum 15612	. 2 ⊢ Σ𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))
19		df-sum 15612	. 2 ⊢ Σ𝑘 ∈ 𝐴 𝐶 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))))
20	17, 18, 19	3eqtr4g 2795	1 ⊢ (∀𝑘 𝐵 = 𝐶 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃wrex 3059  ⦋csb 3848   ⊆ wss 3900  ifcif 4478   class class class wbr 5097   ↦ cmpt 5178  ℩cio 6445  –1-1-onto→wf1o 6490  ‘cfv 6491  (class class class)co 7358  0cc0 11028  1c1 11029   + caddc 11031  ℕcn 12147  ℤcz 12490  ℤ_≥cuz 12753  ...cfz 13425  seqcseq 13926   ⇝ cli 15409  Σcsu 15611

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-xp 5629  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-iota 6447  df-fv 6499  df-ov 7361  df-oprab 7362  df-mpo 7363  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-seq 13927  df-sum 15612

This theorem is referenced by: (None)